Dyck Paths, Configuration Spaces and Polytopes For Linear Nakayama algebras

TL;DR

The paper develops a combinatorial framework for configuration spaces $U_{ ull ext{A}}$ and polytopes $P_{ ull ext{A}}$ associated to quotients $ ull ext{C}A_n/I$ (linear Nakayama algebras) via Dyck paths, establishing a bijection between two-sided ideals and Dyck paths and describing indecomposable modules as grid points below the path. It constructs $U_{ ull ext{A}}$ by $u$-equations on an index set $ ull ext{I}_{ ull ext{A}}$ and represents $U_{ ull ext{A}}$ as an affine open in the toric variety of the polytope $P_{ ull ext{A}}$, with the nonnegative part $(U_{ ull ext{A}})_{ ext{ge}0}$ reflecting the face lattice of $P_{ ull ext{A}}$; a monomial map realizes functorial relations between configuration spaces corresponding to Dyck-path inclusions, while an explicit $F$-polynomial parametrization ties these spaces to toric coordinates. The polytopes $P_{ ull ext{A}}$ are described combinatorially as Minkowski sums of Newton polytopes of indecomposables, shown to be simple, and related to a toric embedding whose rays correspond to the $g$-vectors of $ ull ext{A}$. Overall, the work provides a clear, computable bridge between representation-theoretic data (Dyck paths, ideals, modules) and geometric objects (affine/open varieties, toric embeddings, polytopes) in the linear Nakayama setting, with explicit constructions and examples (e.g., $A_2$ and $A_5$) illustrating the framework.

Abstract

We present a combinatorial model of configuration spaces and polytopes associated to the quotients of $\mathbb{C} A_n$, the path algebra of the linearly oriented $A_n$ quiver, i.e. the algebra of upper triangular matrices. These quotient algebras are known as linear Nakayama algebras. Such configuration spaces were recently introduced for more general algebras by the second author and collaborators. In this special setting, we provide elementary proofs and explicit combinatorial constructions. From a Dyck path we define three related objects: a finite-dimensional algebra, an affine algebraic variety, and a polytope. Moreover, our constructions are natural: each relation in the poset of Dyck paths gives a morphism between the corresponding objects.

Figures (16)

• Figure 1: Dyck path poset for $n=3$.
• Figure 2: The index set $\mathcal{I}_\mathcal{A}$ for $\mathcal{A}=\mathbb{C} A_5/\langle \alpha_1\alpha_2\rangle$ is given by the steps of the Dyck path $D$ (labelled $1,2,3,4,5$ and $\Sigma 1, \Sigma 2, \Sigma 3, \Sigma 4, \Sigma 5$) and the diamonds beneath the Dyck path (labelled $ij$).
• Figure 3: For $\mathcal{A}=\mathbb{C} A_5/\langle \alpha_1\alpha_2\rangle$ and $X = 24 \in \mathcal{I}_\mathcal{A}$, the compatible $Y \in \mathcal{I}_\mathcal{A}$ (for which $c(X,Y) = 0$) are shown in blue, and the incompatible are shown in red.
• Figure 4: For $\mathcal{A}=\mathbb{C} A_5/\langle \alpha_1\alpha_2\rangle$ and $X = 2 \in \mathcal{I}_\mathcal{A}$, the compatible $Y \in \mathcal{I}_\mathcal{A}$ are shown in blue, and the incompatible are shown in red.
• Figure 5: The grid picture for $\mathbb{C} A_2$ (A) can be labelled by the chords of the regular hexagon (B). With this labelling two chords are compatible if and only if they do not cross.

Theorems & Definitions (51)

• Example 1.1
• Example 1.2
• Example 2.1
• Example 2.2
• Proposition 2.3
• proof
• Definition 3.1
• Remark
• Definition 3.2
• Definition 3.3